The purpose of this research study is to determine the long term safety of an investigational drug, B2036-PEG, in patients with acromegaly which is caused by growth hormone secreting tumors. This drug blocks the actions of growth hormone. It may be more effective than the two previously available drugs because they work by different mechanisms which are only effective in selected patients. In contrast, all patients with acromegaly have high growth hormone levels and have growth hormone receptors. Therefore, this drug should have some effect in nearly all patients. Studies performed in animals and humans with this agent showed only a few minor side effects and a decrease in one of the blood proteins (IGF-1) that is responsible for many of the signs and symptoms seen in acromegaly. This study will determine the long term safety and effectiveness of B2036-PEG therapy at different daily doses.